Problem: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}2x+9y &= 1 \\ -6x-9y &= 5\end{align*}$
Answer: Begin by moving the $x$ -term in the second equation to the right side of the equation. $-9y = 6x+5$ Divide both sides by $-9$ to isolate $y$ $y = {-\dfrac{2}{3}x - \dfrac{5}{9}}$ Substitute this expression for $y$ in the first equation. $2x+9({-\dfrac{2}{3}x - \dfrac{5}{9}}) = 1$ $2x - 6x - 5 = 1$ Simplify by combining terms, then solve for $x$ $-4x - 5 = 1$ $-4x = 6$ $x = -\dfrac{3}{2}$ Substitute $-\dfrac{3}{2}$ for $x$ back into the top equation. $2( -\dfrac{3}{2})+9y = 1$ $-3+9y = 1$ $9y = 4$ $y = \dfrac{4}{9}$ The solution is $\enspace x = -\dfrac{3}{2}, \enspace y = \dfrac{4}{9}$.